Question 7.1.5: Using a Table of Conformal Mappings. Use Appendix III to fin......
Using a Table of Conformal Mappings
Use Appendix III to find a conformal mapping from the infinite horizontal strip 0 ≤ y ≤ 2, −∞ < x < ∞, onto the unit disk |w| ≤ 1. Under this mapping, what is the image of the negative x-axis?
Appendix III does not have an entry that maps an infinite horizontal strip onto the unit disk. Therefore, we construct the a conformal mapping that does this by composing two mappings in the table. In Example 4 we found that the infinite horizontal strip 0 ≤ y ≤ 2, −∞ < x < ∞, is mapped onto the upper half-plane by f(z) = e^{πz/2}. In addition, from entry C-4 we see that the upper half-plane is mapped onto the unit disk by g(z)={\frac{i-z}{i+z}}. The composition of these two functions
w=g(f(z))=\frac{i-e^{\pi z/2}}{i+e^{\pi z/2}}therefore maps the strip 0 ≤ y ≤ 2, −∞ < x < ∞, onto the unit disk |w| ≤ 1. Under the first of these successive mappings, the negative real axis is mapped onto the interval (0, 1] in the real axis as was noted in Example 4. Inspection of entry C-4 (or Figure 7.6) reveals that the interval from 0 to C = 1 is mapped onto the circular arc from 1 to C^{\prime} = i on the unit circle |w| = 1. Therefore, we conclude that the negative real axis is mapped onto the circular arc from 1 to i on the unit circle under w={\frac{i-e^{\pi z/2}}{i+e^{\pi z/2}}}.
